Volumes of revolutionEdexcel International A Level Maths: Subtopic test

1. 1
   The region $R$ is bounded by the curve $y=2\sqrt{x}$, the $x$-axis and the lines $x=1$ and $x=4$. The region $R$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.

   (a)

   Which expression gives the volume of the solid?

   [1 mark]

   • A$\pi\int_1^4 2\sqrt{x}\,dx$
   • B$\pi\int_1^4 4x\,dx$
   • C$\pi\int_1^4 2x\,dx$
   • D$\int_1^4 4x\,dx$

   (b)

   Find the volume of the solid.

   [1 mark]

   • A$30$
   • B$15\pi$
   • C$30\pi$
   • D$60\pi$

   (c)

   The line $x=4$ is replaced by the line $x=a$, where $a>1$. The volume of the new solid is $48\pi$. Find the value of $a$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The region $S$ is bounded by the curve $y=e^{2x}$, the coordinate axes and the line $x=1$. The region $S$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.

   (a)

   Which expression gives the volume of the solid?

   [1 mark]

   • A$\pi\int_0^1 e^{2x}\,dx$
   • B$\pi\int_0^1 e^{2x^2}\,dx$
   • C$\pi\int_0^1 4e^{2x}\,dx$
   • D$\pi\int_0^1 e^{4x}\,dx$

   (b)

   Find the exact volume of the solid.

   [1 mark]

   • A$\frac{\pi}{4}(e^4-1)$
   • B$\frac{\pi}{2}(e^4-1)$
   • C$\pi(e^4-1)$
   • D$\frac{\pi}{2}(e^2-1)$

   (c)

   The line $x=1$ is replaced by the line $x=\ln2$. Find the exact volume of the new solid.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The region $R$ is bounded by the curve $y=\frac{1}{\sqrt{2x+1}}$, the coordinate axes and the line $x=4$. The region $R$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.

   (a)

   Show that the volume of the solid is $\pi\ln3$.

   [3 marks]

   (b)

   The line $x=4$ is replaced by the line $x=k$, where $k>0$. Given that the volume of the new solid is $2\pi$, find the exact value of $k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A vase is modelled by rotating the region bounded by the curve $C$ and the $x$-axis through $2\pi$ radians about the $x$-axis. The curve $C$ has parametric equations $x=t^2$, $y=3t(2-t)$ for $0\le t\le2$, with units in centimetres. A calculator may be used.

   (a)

   Find the exact volume of the vase.

   [6 marks]

   (b)

   A designer says that the part of the vase with $x>1$ holds more than twice as much as the part with $0\le x\le1$. Use your integration to evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions